On eigenvalue problems for the random walks on the Sierpinski pre-gaskets

Abstract

We work with increasing finite sets Vm called pre-gaskets approximating the finite Sierpinski gasket located in R N-t (N >_ 3). The eigenvalues of the discrete Laplacian on "Cm under the Dirichlet and Neumann boundary conditions are completely determined using the decimation method due to Rammal.The finite Sierpinski gasket located in the Euclidean N -1 space R N-1 (N >_ 3) is the closure of the union V, of increasing finite sets V,,, (see below for precise definition). We call V,,~ the (m-th step) pre, gasket and we consider a difference operator Hin on Vm corresponding to the simple random walk on Vm.The purpose of this paper is to determine completely the eigenvalues of Hin including their multiplicities under the Dirichlet boundary condigion (Theorem 2.1) and under the Neumann boundary condition (Theorem 3.1). The structure of eigenvalues of H,,, is essentially known owing to the decimation method of RammalToulouse [2] (see also Rammal [3]) relating the eigenvalues of Hin to those of Hin-1 by the quadratic map ~(x) = x(N+2-x). We reproduce the method in Proposition 2.2. Before proving Theorem 2.1 however, we need first to identify the multiplicities of eigenvalues 2, 2N and N + 2, which are excepted in Proposition 2.2. A similar step is necessary in proving Theorem 3.1.Recently Kigami [1] introduced the Laplacian 91 on the Sierpinski gasket as a suitable renormalized limit of Hin as m -* oo. The Dirichlet problem and the Poisson equation associated with A are then solved explicitly. In the present paper, we employ those notations and relations in [1] concerning pre-gaskets. In the forthcoming paper, we shall apply the present result to decide the spectrum and the integrated density of states for Kigami's Laplacian.We now prepare preliminary notions and relations from [1].For Pi E R N-l, i = 1,2,..., N, such that PiPa, j = 2,.. 9 N, are independent, we let i=2 i=29 The aughor would like to thank Professor M. Pukushima and the referee for the mosg helpful suggestions.

The spectrum of infinite regular line graphs

“…Then we obtain a similar result as studied in [7]. It follows from the form of the functions f p ± that the spectrum of pre-n-Sierpinski gasket has Cantor structure as n → ∞.…”

Fractafolds based on the Sierpinski gasket and their spectra

“…In fact, the matching conditions provide necessary and sufficient conditions for gluing together local solutions ∆u i = f i on C i to obtain a global solution ∆u = f (here we assume that the glued functions u and f are continuous). The spectrum of the Laplacian on SG (with Dirichlet or Neumann boundary conditions) was described exactly by Fukushima and Shima [FS], [Sh1] (see also [DSV], [MT], [T] and [GRS] for further elaborations) based on the method of spectral decimation [Sh2]. Our goal is to give an analogous description for fractafolds.…”

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